Geometry

The Pythagorean theorem

The relationship between the sides of a right triangle, why it is true, and how it measures distance.

About 8 minutes

In a right triangle the side opposite the right angle is the hypotenuse; the other two sides are the legs. The Pythagorean theorem links their lengths, and through coordinates it becomes the formula for distance in the plane.

The statement

If the legs have lengths $a$ and $b$ and the hypotenuse has length $c$ , then the square on the hypotenuse has the same area as the two squares on the legs together.

The converse is also true: if the sides of a triangle satisfy the equation, the triangle has a right angle. Builders use this with a 3–4–5 triangle to mark out square corners.

a^{2} + b^{2} = c^{2}

Why it is true

Arrange four copies of the triangle inside a square of side a + b so that they leave a tilted square of side $c$ in the middle. The big square has area (a + b)^2; it is also the four triangles plus the inner square, 4 \cdot \frac12 ab + c^2. Setting these equal and simplifying leaves a^2 + b^2 = c^2.

(a + b)^{2} = 2ab + c^{2}

Distance between two points

The segment joining two points is the hypotenuse of a right triangle whose legs are the horizontal and vertical differences between the points.

d = \sqrt{(x_2 - x_1)^{2} + (y_2 - y_1)^{2}}

Worked examples

The legs of a right triangle are 5 and 12. Find the hypotenuse.

c^2 = 5^2 + 12^2 = 25 + 144 = 169.
c = \sqrt{169} = 13.

Answer: 13

A 10 m ladder reaches 8 m up a wall. How far is its foot from the wall?

The ladder is the hypotenuse: a^2 + 8^2 = 10^2.
a^2 = 100 - 64 = 36.
a = 6.

Answer: 6 m

Common mistakes

Using the theorem on a triangle that has no right angle.
Adding the squares when the unknown is a leg — a leg needs a subtraction: a^2 = c^2 - b^2.
Forgetting the final square root.
Assuming \sqrt{a^2 + b^2} = a + b.

Practice problems

1.Legs 8 and 15. Hypotenuse?
2.Hypotenuse 25, one leg 7. Other leg?
3.Distance from (1, 2) to (4, 6)?
4.Is a triangle with sides 6, 7, 9 right-angled?

Practise this topic with checked answers

Tools for this lesson

Pythagorean Theorem CalculatorFind the missing side of a right triangle from the other two with the Pythagorean theorem, shown step by step.Triangle CalculatorSolve a triangle from SSS, SAS, ASA, AAS or RHS: sides, angles, area by Heron’s formula, heights, medians, bisectors and radii.Distance CalculatorDistance between two points in the plane with the distance formula, shown on a coordinate diagram.

Next lesson: The unit circle and trigonometric functions

The statement

If the legs have lengths

a

and

b

and the hypotenuse has length

c

, then the square on the hypotenuse has the same area as the two squares on the legs together.

The converse is also true: if the sides of a triangle satisfy the equation, the triangle has a right angle. Builders use this with a 3–4–5 triangle to mark out square corners.

a^{2} + b^{2} = c^{2}

a^{2} + b^{2} = c^{2}

Why it is true

Arrange four copies of the triangle inside a square of side

a + b

a + b so that they leave a tilted square of side

c

in the middle. The big square has area

(a + b)^2

(a + b)^2; it is also the four triangles plus the inner square,

4 \cdot \frac12 ab + c^2

4 \cdot \frac12 ab + c^2. Setting these equal and simplifying leaves

a^2 + b^2 = c^2

a^2 + b^2 = c^2.

(a + b)^{2} = 2ab + c^{2}